Similar Grandi's series, Divergence of the sum of the reciprocals of the primes, Riemann zeta function | ||
In mathematics, an infinite arithmetic series is an infinite series whose terms are in an arithmetic progression. Examples are 1 + 1 + 1 + 1 + · · · and 1 + 2 + 3 + 4 + · · ·. The general form for an infinite arithmetic series is
If a = b = 0, then the sum of the series is 0. If either a or b is nonzero while the other is, then the series diverges and has no sum in the usual sense.
Zeta regularization
The zeta-regularized sum of an arithmetic series of the right form is a value of the associated Hurwitz zeta function,
Although zeta regularization sums 1 + 1 + 1 + 1 + · · · to ζR(0) = −1/2 and 1 + 2 + 3 + 4 + · · · to ζR(−1) = −1/12, where ζ is the Riemann zeta function, the above form is not in general equal to
References
Infinite arithmetic series Wikipedia(Text) CC BY-SA